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Question 1209616: Suppose the domain of f is (-1,3). Define the function g by
g(x) = f((x + 1)(x - 2)).
What is the domain of g?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to determine the domain of g:
**1. Understand the relationship between f and g:**
The function g(x) is defined in terms of f. The input to f is the expression (x + 1)(x - 2). For g(x) to be defined, this expression must fall within the domain of f, which is (-1, 3).
**2. Set up the inequality:**
We need to find the values of x for which:
-1 < (x + 1)(x - 2) < 3
**3. Solve the inequalities:**
We'll solve this as two separate inequalities:
* **-1 < (x + 1)(x - 2)**
-1 < x² - x - 2
0 < x² - x - 1
To solve this, we find the roots of the quadratic x² - x - 1 = 0 using the quadratic formula:
x = (1 ± √(1 + 4))/2 = (1 ± √5)/2
So, x < (1 - √5)/2 or x > (1 + √5)/2
* **(x + 1)(x - 2) < 3**
x² - x - 2 < 3
x² - x - 5 < 0
Again, find the roots of x² - x - 5 = 0:
x = (1 ± √(1 + 20))/2 = (1 ± √21)/2
So, (1 - √21)/2 < x < (1 + √21)/2
**4. Combine the solutions:**
We need to find where *both* inequalities are true. Let's approximate the roots:
* (1 - √5)/2 ≈ -0.618
* (1 + √5)/2 ≈ 1.618
* (1 - √21)/2 ≈ -1.791
* (1 + √21)/2 ≈ 2.791
Combining the inequalities, we get:
(1 - √21)/2 < x < (1 - √5)/2 *or* (1 + √5)/2 < x < (1 + √21)/2
**5. State the domain of g:**
The domain of g is the union of the two intervals:
((1 - √21)/2, (1 - √5)/2) ∪ ((1 + √5)/2, (1 + √21)/2)
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